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Implicit function

A function given by an equation , where , , , , and , and are certain sets, i.e. a function such that for any . If , and are topological spaces and if for some point , then under certain conditions the equation is uniquely solvable in one of the variables in some neighbourhood of . Properties of the solution of this equation are described by implicit-function theorems.

The simplest implicit-function theorem is as follows. Suppose that and are subsets of the real line , let , , and let be an interior point of the plane set ; if is continuous in some neighbourhood of , if and if there are a and an such that , for any fixed , is strictly monotone on as a function of , then there is a such that there is a unique function

for which for all ; moreover, is continuous and .

Figure: i050310a

The hypotheses of this theorem are satisfied if is continuous in a neighbourhood of , if the partial derivative exists and is continuous at , if , and if . If in addition the partial derivative exists and is continuous at , then the implicit function is differentiable at , and

This theorem has been generalized to the case of a system of equations, that is, when is a vector function. Let and be - and -dimensional Euclidean spaces with fixed coordinate systems and points and , respectively. Suppose that maps a certain neighbourhood of (, ) into and that , , are the coordinate functions (of the variables ) of , that is, . If is differentiable on , if and if the Jacobian

then there are neighbourhoods and of and , respectively, , and a unique mapping such that for all . Here , is differentiable on , and if , then the explicit expression for the partial derivatives , , , can be found from the system of linear equations in these derivatives:

, is fixed . Sometimes the main assertion of the theorem is stated as follows: There are neighbourhoods of in and of in , , and a unique mapping such that and for all . In other words, the conditions

are equivalent to , . In this case one says that the equation is uniquely solvable in the neighbourhood of .

The classical implicit-function theorem thus stated generalizes to the case of more general spaces in the following manner. Let be a topological space, let and be affine normed spaces over the field of real or complex numbers, that is, affine spaces over the relevant field to which are associated normed vector spaces and , being complete, let be the set of continuous linear mappings from into , and let be an open set in the product space , , , .

Let be a continuous mapping and . If for every fixed and the mapping has a partial Fréchet derivative, if is a continuous mapping and if the linear mapping has a continuous inverse linear mapping (that is, it is an invertible element of ), then there exist open sets and , , , such that for any there is a unique element , denoted by , satisfying the equations

The function thus defined is a continuous mapping from into , and .

If is also an affine normed space, then under certain conditions the implicit function which satisfies the equation

(1)

is also differentiable. Namely, let , and be affine normed spaces, let be an open set in , let , , , and let be the implicit mapping given by (1), taking a certain neighbourhood of into an open subset of , . Thus, for all ,

(2)

Suppose also that is continuous at and that . If is differentiable at , if its partial Fréchet derivatives and are continuous linear operators taking the vector spaces and associated with and into the vector space associated with , and if the operator is an invertible element of , then is differentiable at and its Fréchet derivative is given by

This is obtained as a result of formally differentiating (2):

and multiplying this equality on the left by .

If in addition the mapping is continuously differentiable on , if the implicit function is continuous on , , and if for any the partial Fréchet derivative is an invertible element of , then is a continuously-differentiable mapping of into .

In the general case one can also indicate conditions for the existence and the uniqueness of the implicit function in terms of the continuity of the Fréchet derivative: If is complete, if the mapping is continuously differentiable on , if , and if the partial Fréchet derivative is an invertible element of , then (1) is uniquely solvable in a sufficiently small neighbourhood of , i.e. there exist neighbourhoods of in and of in , , and a unique implicit function satisfying (2). Here is also continuously differentiable on . In this form the implicit-function theorem for normed spaces is a direct generalization of the corresponding classic implicit-function theorem for a single scalar equation in two variables.

Furthermore, if is a -times continuously-differentiable mapping in a neighbourhood of , then the implicit function is also times continuously differentiable.